Firstly, it is important to be aware that our understanding of the civilisations and culture of Pre-Columbian America is far from complete. Secondly, I don't propose to attempt to cover every civilisation, so this will - at best - be only a partial answer.
However, with those caveats:
As far as I am aware, we have no evidence that any pre-Columbian cultures or civilisations knew the true shape of the Earth. All the evidence we do have supports the idea that they all supported a flat-Earth model. What is more, there seems to be a considerable degree of overlap in the cosmologies of the major civilisations.
The Aztecs believed that the Earth had the approximate shape of a giant disk. That disk was divided into four cardinal directions. Above the Earth were 13 heavens, while below the Earth were the 'nine Hells of Mictlan'.
The Mayans also believed that 13 heavens were stacked in layers above the Earth. The Earth itself was flat and four-cornered, and carried on the back of a giant reptile (probably a crocodile) which was, in turn,floating on the ocean. Below the Earth were nine underworlds once again.
Olmec, Zapotec, Mextec:
In the paper, Cosmology in Mesoamerica, Keith Jordan argues that:
While each distinct culture generated its own cosmology, all such models of the nature of the cosmos are united by common features marking them as distinctively Mesoamerican.
Among those 'common features' seem to be the idea of 13 heavens above a flat Earth, with nine underworlds below.
Our current understanding is that this represents part of a continuity of belief which extended from the Olmec civilisation (c 1500 BCE - c 400 BCE) until the arrival of the Spanish, although there remains some debate about just how much the beliefs of the Olmec influenced later civilisations. See, for example, The gods and symbols of ancient Mexico and the Maya by Miller & Taube (Thames & Hudson, 1993).
The Olmec civilisation is the oldest of these groups, and evidence for their world view in this regard was found on a square jade plate excavated at Ahuelican in State of Guerrero:
"On its small surface were recorded the three regions of the Mesoamerican universe. At the top, the sky with thirteen estrata surrounding a glyph in form of X that for the Olmecs and Mayas represented the sky. Immediately below appears a quincunx with four elements similar to seeds representing the directions of the universe or the solstitial extremes in the horizon. In the center was carved a fruiting maize plant with three cobs. This plant arises of a design that seems to be an architectural structure that stands on a hill. The base of this hill leads to the underworld, to the primeval water indicated by three oval elements and one rectangular."
For more on our current understanding of the relationships between the various early Mesoamerican groups, see Jeffrey P Blomster: Complexity, interaction, and epistemology: Mixtecs, Zapotecs, and Olmecs in Early Formative Mesoamerica, Ancient Mesoamerica, Vol 21, No 1 (Spring 2010), pp 135-149.
The Inca called their land "Tahuantinsuyu" which (according to the course notes for Lecture 4 of Astronomy 161 at Ohio State) means "The Four Quarters of the Earth". This suggests that they had similar views to those of the Aztec and the Maya. In his paper Animals and Astronomy in the Quechua Universe, Gary Urton observed:
"... the celestial River is believed to carry into the sky the actual water which flows through the Vilcanota River. As the Vilcanota River flows from the southeast to the northwest, it carries terrestrial water to the edge of the earth. The water then flows into the mar, the cosmic sea, which completely encircles the earth.
Again, this suggests that the Inca also believed that the Earth was flat and four-cornered.